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Video: Division Algorithm For Polynomials Video Lecture - Class 10
FAQs on Video: Division Algorithm For Polynomials Video Lecture - Class 10
| 1. What is the division algorithm for polynomials? |  |
Ans. The division algorithm for polynomials states that for any two polynomials, dividend and divisor, there exists a unique quotient and remainder such that when the dividend is divided by the divisor, the remainder is either zero or has a degree lower than the divisor.
| 2. How is the division algorithm applied to polynomials? | |
Ans. To apply the division algorithm to polynomials, we divide the dividend by the divisor using long division or synthetic division method. The resulting quotient and remainder satisfy the properties mentioned in the algorithm.
| 3. Can you explain the steps involved in the division algorithm for polynomials? | |
Ans. Sure! The steps involved in the division algorithm for polynomials are as follows:
1. Divide the first term of the dividend by the first term of the divisor to obtain the first term of the quotient.
2. Multiply the entire divisor by the first term of the quotient.
3. Subtract the obtained product from the dividend.
4. Repeat steps 1-3 with the new dividend (obtained in step 3) until the degree of the new dividend is lower than the divisor.
5. The final quotient obtained after all the divisions is the quotient of the polynomial division.
6. The remaining polynomial, if any, after division is the remainder.
| 4. Is the remainder always zero in polynomial division? | |
Ans. No, the remainder is not always zero in polynomial division. According to the division algorithm for polynomials, the remainder can be either zero or have a degree lower than the divisor. If the remainder is zero, it means that the divisor is a factor of the dividend. If the remainder is non-zero, it indicates that there is no exact division and there is a leftover polynomial.
| 5. Can the division algorithm for polynomials be used to factorize polynomials? | |
Ans. Yes, the division algorithm for polynomials can be used to factorize polynomials. By dividing a polynomial by its factors, we can determine the quotient and remainder. If the remainder is zero, it implies that the divisor is a factor of the polynomial. Therefore, the division algorithm helps in identifying the factors of a polynomial and can aid in the process of factorization.
Text Transcript from Video
division algorithm for polynomials
divide 2x cubed plus 3x squared minus X
plus 5 by 1 minus X plus x squared we
first arrange the terms of the dividend
and the divisor in the standard form
that is we write them in the decreasing
order of the degrees here the dividend
is already in the standard form we
rewrite the divisor in the standard form
as x squared minus X plus 1 we start the
division process step 1 to obtain the
first term of the quotient divide the
highest degree term of the dividend that
is 2x cubed by the highest degree term
of the divisor that is x squared we get
2x
now multiply each term of the divisor by
2x we get 2x cubed minus 2x squared plus
2x subtracting we get the remainder 5x
squared minus 3x plus 5 step 2 to obtain
the second term of the quotient divide
the highest degree term of the new
dividend that is 5x squared by the
highest degree term of the divisor that
is x squared we get +5 now multiply each
term of the divisor by +5 we get 5x
squared minus 5x plus 5 subtracting we
get the remainder 2x step 3 check that
the degree of the remainder is less than
the degree of the divisor the degree of
the remainder is 1 and the degree of the
divisor is 2 so we cannot continue the
division any further the cogent is 2x
plus 5 and the remainder is 2x
we can verify the
in this way the divisor is x squared
minus X plus 1 the coefficient is 2x
plus 5
divisor
is equal to x squared minus X plus 1
into 2 X plus 5 that is equal to
you
to X cubed minus two x squared plus two
x plus five x squared minus five x plus
five divided into cogent plus remainder
is equal to two X cubed minus 2x squared
plus 2x plus five x squared minus five x
plus five plus two x that is equal to
two X cubed plus three x squared minus X
plus 5 that is equal to the dividend
dividend is equal to divisor into
Koecher plus remainder
in general if px and GX are any two
polynomials with GX not equal to 0 then
we can find polynomials QX and rx such
that px is equal to GX into QX plus rx
where Rx is equal to 0 or the degree of
our X is less than the degree of GX this
result is known as division algorithm
for polynomials
divide 2x cubed plus 3x squared minus X
plus 5 by 1 minus X plus x squared we
first arrange the terms of the dividend
and the divisor in the standard form
that is we write them in the decreasing
order of the degrees here the dividend
is already in the standard form we
rewrite the divisor in the standard form
as x squared minus X plus 1 we start the
division process step 1 to obtain the
first term of the quotient divide the
highest degree term of the dividend that
is 2x cubed by the highest degree term
of the divisor that is x squared we get
2x
now multiply each term of the divisor by
2x we get 2x cubed minus 2x squared plus
2x subtracting we get the remainder 5x
squared minus 3x plus 5 step 2 to obtain
the second term of the quotient divide
the highest degree term of the new
dividend that is 5x squared by the
highest degree term of the divisor that
is x squared we get +5 now multiply each
term of the divisor by +5 we get 5x
squared minus 5x plus 5 subtracting we
get the remainder 2x step 3 check that
the degree of the remainder is less than
the degree of the divisor the degree of
the remainder is 1 and the degree of the
divisor is 2 so we cannot continue the
division any further the cogent is 2x
plus 5 and the remainder is 2x
we can verify the
in this way the divisor is x squared
minus X plus 1 the coefficient is 2x
plus 5
divisor
is equal to x squared minus X plus 1
into 2 X plus 5 that is equal to
you
to X cubed minus two x squared plus two
x plus five x squared minus five x plus
five divided into cogent plus remainder
is equal to two X cubed minus 2x squared
plus 2x plus five x squared minus five x
plus five plus two x that is equal to
two X cubed plus three x squared minus X
plus 5 that is equal to the dividend
dividend is equal to divisor into
Koecher plus remainder
in general if px and GX are any two
polynomials with GX not equal to 0 then
we can find polynomials QX and rx such
that px is equal to GX into QX plus rx
where Rx is equal to 0 or the degree of
our X is less than the degree of GX this
result is known as division algorithm
for polynomials
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